Kinetic Energy and Molecular Speeds

Why gases have a distribution of speeds, not one speed

Gas molecules are in ceaseless motion, constantly colliding with each other and with the container walls. Every collision can change a molecule's speed and redistribute energy between the colliding pair. As a result, at any instant, not all molecules move at the same speed — some are fast, some slow, most somewhere in between. All we can meaningfully talk about is an average.

For $n$ molecules with individual speeds $u_1, u_2, \ldots, u_n$, the average speed is:

The Maxwell-Boltzmann distribution

Maxwell and Boltzmann showed that the actual spread of molecular speeds depends on both temperature and molecular mass. Maxwell derived a formula giving the number of molecules possessing any particular speed; Fig. 5.8 sketches this — the number of molecules plotted against speed, at two temperatures $T_1$ and $T_2$ (with $T_2 > T_1$) — known as the Maxwell-Boltzmann distribution of speeds.

A few features of this curve:

• Very few molecules move extremely slowly or extremely fast — the bulk cluster near the middle.
• The peak of the curve marks the speed possessed by the largest number of molecules — the most probable speed, $u_{mp}$, which is close to (but not identical to) the average speed.
• Raising the temperature shifts $u_{mp}$ higher and broadens the curve — more molecules now move at higher speeds.
• Heavier molecules move more slowly than lighter ones at the same temperature. Fig. 5.9 compares nitrogen and chlorine: at any given temperature, the lighter N$_2$ molecules have a higher most-probable speed than the heavier Cl$_2$ molecules.

Even though any individual molecule's speed keeps changing from collision to collision, the overall distribution of speeds at a fixed temperature stays the same.

From speed to kinetic energy

A particle's kinetic energy is $\dfrac{1}{2}mu^{2}$. To get the average translational kinetic energy, $\dfrac{1}{2}m\overline{u^{2}}$, we first need the mean of the squares of all the individual speeds:

$\overline{u^{2}} = \frac{u_{1}^{2}+u_{2}^{2}+\cdots+u_{n}^{2}}{n}$ …